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Theorem oppglsm 17880
 Description: The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
oppglsm.o 𝑂 = (oppg𝐺)
oppglsm.p = (LSSum‘𝐺)
Assertion
Ref Expression
oppglsm (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇)

Proof of Theorem oppglsm
Dummy variables 𝑢 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2610 . . . . . . . 8 (+g𝐺) = (+g𝐺)
3 oppglsm.p . . . . . . . 8 = (LSSum‘𝐺)
41, 2, 3lsmfval 17876 . . . . . . 7 (𝐺 ∈ V → = (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))))
54tposeqd 7242 . . . . . 6 (𝐺 ∈ V → tpos = tpos (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))))
6 eqid 2610 . . . . . . . . . . . . 13 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
76reldmmpt2 6669 . . . . . . . . . . . 12 Rel dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
86mpt2fun 6660 . . . . . . . . . . . . 13 Fun (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
9 funforn 6035 . . . . . . . . . . . . 13 (Fun (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) ↔ (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)))
108, 9mpbi 219 . . . . . . . . . . . 12 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
11 tposfo2 7262 . . . . . . . . . . . 12 (Rel dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) → ((𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) → tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))))
127, 10, 11mp2 9 . . . . . . . . . . 11 tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
13 forn 6031 . . . . . . . . . . 11 (tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)):dom (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))–onto→ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) → ran tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)))
1412, 13ax-mp 5 . . . . . . . . . 10 ran tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))
15 oppglsm.o . . . . . . . . . . . . . . . 16 𝑂 = (oppg𝐺)
16 eqid 2610 . . . . . . . . . . . . . . . 16 (+g𝑂) = (+g𝑂)
172, 15, 16oppgplus 17602 . . . . . . . . . . . . . . 15 (𝑥(+g𝑂)𝑦) = (𝑦(+g𝐺)𝑥)
1817eqcomi 2619 . . . . . . . . . . . . . 14 (𝑦(+g𝐺)𝑥) = (𝑥(+g𝑂)𝑦)
1918a1i 11 . . . . . . . . . . . . 13 ((𝑦𝑢𝑥𝑡) → (𝑦(+g𝐺)𝑥) = (𝑥(+g𝑂)𝑦))
2019mpt2eq3ia 6618 . . . . . . . . . . . 12 (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑦𝑢, 𝑥𝑡 ↦ (𝑥(+g𝑂)𝑦))
2120tposmpt2 7276 . . . . . . . . . . 11 tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))
2221rneqi 5273 . . . . . . . . . 10 ran tpos (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))
2314, 22eqtr3i 2634 . . . . . . . . 9 ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))
2423a1i 11 . . . . . . . 8 ((𝑢 ∈ 𝒫 (Base‘𝐺) ∧ 𝑡 ∈ 𝒫 (Base‘𝐺)) → ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
2524mpt2eq3ia 6618 . . . . . . 7 (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))) = (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
2625tposmpt2 7276 . . . . . 6 tpos (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦𝑢, 𝑥𝑡 ↦ (𝑦(+g𝐺)𝑥))) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
275, 26syl6eq 2660 . . . . 5 (𝐺 ∈ V → tpos = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))))
28 fvex 6113 . . . . . . 7 (oppg𝐺) ∈ V
2915, 28eqeltri 2684 . . . . . 6 𝑂 ∈ V
3015, 1oppgbas 17604 . . . . . . 7 (Base‘𝐺) = (Base‘𝑂)
31 eqid 2610 . . . . . . 7 (LSSum‘𝑂) = (LSSum‘𝑂)
3230, 16, 31lsmfval 17876 . . . . . 6 (𝑂 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))))
3329, 32ax-mp 5 . . . . 5 (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
3427, 33syl6reqr 2663 . . . 4 (𝐺 ∈ V → (LSSum‘𝑂) = tpos )
3534oveqd 6566 . . 3 (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇tpos 𝑈))
36 ovtpos 7254 . . 3 (𝑇tpos 𝑈) = (𝑈 𝑇)
3735, 36syl6eq 2660 . 2 (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇))
38 eqid 2610 . . . . . . 7 (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)
39 0ex 4718 . . . . . . 7 ∅ ∈ V
40 eqidd 2611 . . . . . . 7 ((𝑡 = 𝑇𝑢 = 𝑈) → ∅ = ∅)
4138, 39, 40elovmpt2 6777 . . . . . 6 (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ↔ (𝑇 ∈ 𝒫 (Base‘𝐺) ∧ 𝑈 ∈ 𝒫 (Base‘𝐺) ∧ 𝑥 ∈ ∅))
4241simp3bi 1071 . . . . 5 (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) → 𝑥 ∈ ∅)
4342ssriv 3572 . . . 4 (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ⊆ ∅
44 ss0 3926 . . . 4 ((𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ⊆ ∅ → (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) = ∅)
4543, 44ax-mp 5 . . 3 (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) = ∅
46 elpwi 4117 . . . . . . . . . . . . 13 (𝑡 ∈ 𝒫 (Base‘𝐺) → 𝑡 ⊆ (Base‘𝐺))
47463ad2ant2 1076 . . . . . . . . . . . 12 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ (Base‘𝐺))
48 fvprc 6097 . . . . . . . . . . . . 13 𝐺 ∈ V → (Base‘𝐺) = ∅)
49483ad2ant1 1075 . . . . . . . . . . . 12 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (Base‘𝐺) = ∅)
5047, 49sseqtrd 3604 . . . . . . . . . . 11 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ ∅)
51 ss0 3926 . . . . . . . . . . 11 (𝑡 ⊆ ∅ → 𝑡 = ∅)
5250, 51syl 17 . . . . . . . . . 10 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 = ∅)
53 eqid 2610 . . . . . . . . . 10 𝑢 = 𝑢
54 mpt2eq12 6613 . . . . . . . . . 10 ((𝑡 = ∅ ∧ 𝑢 = 𝑢) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = (𝑥 ∈ ∅, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
5552, 53, 54sylancl 693 . . . . . . . . 9 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = (𝑥 ∈ ∅, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)))
56 mpt20 6623 . . . . . . . . 9 (𝑥 ∈ ∅, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ∅
5755, 56syl6eq 2660 . . . . . . . 8 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ∅)
5857rneqd 5274 . . . . . . 7 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ran ∅)
59 rn0 5298 . . . . . . 7 ran ∅ = ∅
6058, 59syl6eq 2660 . . . . . 6 ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦)) = ∅)
6160mpt2eq3dva 6617 . . . . 5 𝐺 ∈ V → (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑂)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅))
6233, 61syl5eq 2656 . . . 4 𝐺 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅))
6362oveqd 6566 . . 3 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈))
64 fvprc 6097 . . . . . 6 𝐺 ∈ V → (LSSum‘𝐺) = ∅)
653, 64syl5eq 2656 . . . . 5 𝐺 ∈ V → = ∅)
6665oveqd 6566 . . . 4 𝐺 ∈ V → (𝑈 𝑇) = (𝑈𝑇))
67 0ov 6580 . . . 4 (𝑈𝑇) = ∅
6866, 67syl6eq 2660 . . 3 𝐺 ∈ V → (𝑈 𝑇) = ∅)
6945, 63, 683eqtr4a 2670 . 2 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇))
7037, 69pm2.61i 175 1 (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ◡ccnv 5037  dom cdm 5038  ran crn 5039  Rel wrel 5043  Fun wfun 5798  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  tpos ctpos 7238  Basecbs 15695  +gcplusg 15768  oppgcoppg 17598  LSSumclsm 17872 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-oppg 17599  df-lsm 17874 This theorem is referenced by:  lsmmod2  17912  lsmdisj2r  17921
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